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Theorem rspceaov 27928
Description: A frequently used special case of rspc2ev 3020 for operation values, analogous to rspceov 6075. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
rspceaov  |-  ( ( C  e.  A  /\  D  e.  B  /\  S  = (( C F D))  )  ->  E. x  e.  A  E. y  e.  B  S  = (( x F y))  )
Distinct variable groups:    x, A    x, y, B    x, C, y    y, D    x, F, y    x, S, y
Allowed substitution hints:    A( y)    D( x)

Proof of Theorem rspceaov
StepHypRef Expression
1 eqidd 2405 . . . 4  |-  ( x  =  C  ->  F  =  F )
2 id 20 . . . 4  |-  ( x  =  C  ->  x  =  C )
3 eqidd 2405 . . . 4  |-  ( x  =  C  ->  y  =  y )
41, 2, 3aoveq123d 27909 . . 3  |-  ( x  =  C  -> (( x F y))  = (( C F y))  )
54eqeq2d 2415 . 2  |-  ( x  =  C  ->  ( S  = (( x F
y)) 
<->  S  = (( C F y))  ) )
6 eqidd 2405 . . . 4  |-  ( y  =  D  ->  F  =  F )
7 eqidd 2405 . . . 4  |-  ( y  =  D  ->  C  =  C )
8 id 20 . . . 4  |-  ( y  =  D  ->  y  =  D )
96, 7, 8aoveq123d 27909 . . 3  |-  ( y  =  D  -> (( C F y))  = (( C F D))  )
109eqeq2d 2415 . 2  |-  ( y  =  D  ->  ( S  = (( C F
y)) 
<->  S  = (( C F D))  ) )
115, 10rspc2ev 3020 1  |-  ( ( C  e.  A  /\  D  e.  B  /\  S  = (( C F D))  )  ->  E. x  e.  A  E. y  e.  B  S  = (( x F y))  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667   ((caov 27840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-res 4849  df-iota 5377  df-fun 5415  df-fv 5421  df-dfat 27841  df-afv 27842  df-aov 27843
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